You will not be able to activate the free trial on the same Apple ID. Apple’s
terms and conditions are very clear on this:
That said, Apple also says that it’s one offer per
family, yet some on this thread have reported that they’ve been able to get another trial by using a different Apple ID in the same family sharing group, as long as the original family subscription has been cancelled first. This isn’t supposed to work, but YMMV.
If you use a completely different Apple ID, not associated with the original one, you should be able to redeem the offer. You may need to ensure that this is the Apple ID you initially activate your new Apple TV with in order to qualify, however, and of course you’ll need to switch to this Apple ID whenever you want to watch Apple TV+. This is fairly easy to do on the Apple TV, but it’s more of a pain on the iPhone and iPad.